Theorem eceq2i 6735

Description: Equality theorem for the A-coset and B-coset of C, inference version. (Contributed by Peter Mazsa, 11-May-2021.)

Hypothesis

Ref	Expression
eceq2i.1	|- A = B

Assertion

Ref	Expression
eceq2i	|- [ C ] A = [ C ] B

Proof of Theorem eceq2i

Step	Hyp	Ref	Expression
1		eceq2i.1	. 2 |- A = B
2		eceq2 6734	. 2 |- ( A = B -> [ C ] A = [ C ] B )
3	1, 2	ax-mp 5	1 |- [ C ] A = [ C ] B

Syntax hints:   = wceq 1395   [cec 6695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-ec 6699

This theorem is referenced by:  ecqusaddd  13815